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Theorem ndmovg 7589
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
Assertion
Ref Expression
ndmovg ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ndmovg
StepHypRef Expression
1 df-ov 7411 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 eleq2 2822 . . . . . 6 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)))
3 opelxp 5712 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
42, 3bitrdi 286 . . . . 5 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑅𝐵𝑆)))
54notbid 317 . . . 4 (dom 𝐹 = (𝑅 × 𝑆) → (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ¬ (𝐴𝑅𝐵𝑆)))
6 ndmfv 6926 . . . 4 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6syl6bir 253 . . 3 (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴𝑅𝐵𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅))
87imp 407 . 2 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
91, 8eqtrid 2784 1 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  c0 4322  cop 4634   × cxp 5674  dom cdm 5676  cfv 6543  (class class class)co 7408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  ndmov  7590  curry1val  8090  curry2val  8094  1div0  11872  repsundef  14720  cshnz  14741  mamufacex  21890  mavmulsolcl  22052  mavmul0g  22054  iscau2  24793  1div0apr  29718  rrxsphere  47424
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